Cross Sections

Abstract

This lesson utilizes the concepts of cross-sections of three-dimensional models to demonstrate the derivation of two-dimensional shapes.

Objectives

Upon completion of this lesson, students will:

  • understand the concept of cross-sections
  • gain experience manipulating polygons, ellipses, parabolas, and hyperbolas
  • learn the difference between ellipses, parabolas, hyperbolas, and circles as they relate to conic sections
  • gain an intuitive understanding of the relationship between cross sections of three-dimensional objects and two-dimensional figures
  • discover the relationships between the number of faces of a three-dimensional figure and its two-dimensional cross-sections

Standards Addressed

Grade 9

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student solves problems (including real-world situations).
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 10

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student solves problems (including real-world situations).
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Geometry

  • Expressing Geometric Properties with Equations

    • Translate between the geometric description and the equation for a conic section

  • Geometric Measurement and Dimension

    • Visualize relationships between two-dimensional and three- dimensional objects

  • Modeling with Geometry

    • Apply geometric concepts in modeling situations

Grades 9-12

  • Geometry

    • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
    • Use visualization, spatial reasoning, and geometric modeling to solve problems

Geometry

  • Geometry and Measurement

    • Competency Goal 2: The learner will use geometric and algebraic properties of figures to solve problems and write proofs.

Geometry

  • Algebra

    • Students will recognize, extend, create, and analyze a variety of geometric, spatial, and numerical patterns; solve real-world problems related to algebra and geometry; and use properties of various geometric figures to analyze and solve problems.

  • Geometry

    • Students will investigate, model, and apply geometric properties and relationships and use indirect reasoning to make conjectures; deductive reasoning to draw conclusions; and both inductive and deductive reasoning to establish the truth of statements.

Geometry

  • Dimensionality and the Geometry of Location

    • 6. The student analyzes the relationship between three- dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems.
    • 7. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly.

  • Geometric Patterns

    • 5. The student uses a variety of representations to describe geometric relationships and solve problems.

  • Geometric Structure

    • 4. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems.

Secondary

  • Geometry

    • G.12 The student will make a model of a three-dimensional figure from a two-dimensional drawing and make a two-dimensional representation of a three-dimensional object. Models and representations will include scale drawings, perspective drawings, blueprints, or computer simulations.

Student Prerequisites

  • Geometry: Students must be able to:
    • identify and describe two-dimensional figures
    • identify and describe three-dimensional objects
  • Algebra: Students must be able to:
    • work with two-dimensional graphs
  • Technology: Students must be able to:
    • perform basic mouse manipulations such as point, click, and drag
    • use a browser for experimenting with the activities

Teacher Preparation

  • Access to a browser
  • A copy of the worksheet for each student

Key Terms

coordinate plane

A plane with a point selected as an origin, some length selected as a unit of distance, and two perpendicular lines that intersect at the origin, with positive and negative direction selected on each line. Traditionally, the lines are called x (drawn from left to right, with positive direction to the right of the origin) and y (drawn from bottom to top, with positive direction upward of the origin). Coordinates of a point are determined by the distance of this point from the lines, and the signs of the coordinates are determined by whether the point is in the positive or in the negative direction from the origin

graph

A visual representation of data that displays the relationship among variables, usually cast along x and y axes.

polygon

A closed plane figure formed by three or more line segments that do not cross over each other

polyhedra

Any solid figure with an outer surface composed of polygon faces

rigid motion

A rigid motion, of the plane or of space, is one that keeps the distances between all pairs of points unchanged. Rotations, reflections and translations are examples of rigid motions.

rotate

To perform a rotation

rotation

A rotation in the plane is a rigid motion keeping exactly one point fixed, called the "center" of the rotation. Since distances are unchanged, all the other points can be thought of as having moved on circles whose center is the center of the rotation. The "angle" of the rotation is how far around the circles the points travel. A rotation in three-dimensional space is a rigid motion that keeps the points on one line fixed, called the "axis" of the rotation, with the rest of the points moving some constant angle around circles centered on and perpendicular to the axis.

Lesson Outline

  1. Focus and Review

    Ask the following opening questions:

    • If you place a cone on the table and cut a slice that is parallel to the table, what will that face look like?
      • Use a styrofoam cone, if helpful for the students
      • Ask the students to sketch what it might look like
      • After students sketch the resulting cross section, slice the cone to see if they are correct.
    • As a class, discuss how you can predict what a particular cross section will look like.
      • Have students explain their method and why it should work.
      • Ask students if their methods would work for non-cone objects like prisms or pyramids.

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, we are going to learn how to find the two-dimensional cross section of a three-dimensional object.
    • We are going to use computers to visualize these cross sections, but please do not turn your computers on or go to this website until I ask you to. I want to show you a little about this first.

  3. Teacher Input

    • Introduce cross sections by leading the class in a discussion.
    • If the students have trouble with the concept, demonstrate with styrofoam objects.

  4. Guided Practice

    • Bring up the Cross Section Flyer activity on an overhead projector to demonstrate how the activity works.
    • Explain that the diagram on the left side shows a three-dimensional object that is being "sliced" by a two-dimensional plane to form a cross-section.
    • Explain how the cross section is then shown in two dimensions in the graph on the right side.
    • Walk the students through the applet, showing how to move the slicing plane and how changes in the slicing plane affect the cross section shown on both the three-dimensional object and the two-dimensional graph.

  5. Independent Practice

  6. Closure

    As a class, discuss the findings of the lesson:

    • Review the questions on the worksheet and compare results.
    • Ask students if they now know a better way to predict what a particular cross section will look like.

Alternate Outline

If only one computer is available for the classroom, this lesson can be rearranged in the following way:

  • The teacher may do this activity as a demonstration.
  • The class can work together to answer the questions on the worksheet, with one student controlling the applet on a projector for everyone to see.

Suggested Follow-Up

Students who understand the basic concept of cross sections can explore algebraic equations of conic sections using the Conic Section Flyer.